Lipschitz Continuous Function Map Null Set to Null Set

Lipschitz Continuity

Using the Lipschitz continuity of the regressors the proof is straightforward, hence we omit the details.

From: Handbook of Statistics , 2012

Handbook of Dynamical Systems

Boris Hasselblatt , Yakov Pesin , in Handbook of Dynamical Systems, 2006

Theorem 8.1

A volume-preserving essentially accessible dynamically coherent partially hyperbolic diffeomorphism is ergodic if it has Lipschitz continuous center foliation and Lipschitz continuous stable and unstable holonomy maps between center transversals.

Proof. (8.2)

follows from the conditions of the theorem.

The assumptions on Lipschitz continuity are very strong and "typically" fail (see Section 6.2). On one hand, the modern work in the field has found ways to circumvent the requirement that the center foliation be Lipschitz continuous, and on the other hand, in the presence of dynamical coherence, Lipschitz continuity of the holonomies between center transversals is obtained from the following condition.

Definition 8.2 [37]

We say that f is center-bunched if max $\left\{{\mu }_1,{\lambda }_3^{-1}\right\}<{\lambda }_2/{\mu }_2$ . in (2.7)

This definition due to Burns and Wilkinson imposes a much weaker constraint than earlier versions; in fact, their results assume an even weaker condition one might call "pointwise center bunching": $\left\{{\mu }_1\left(p\right),{\lambda }_3^{-1}\left(p\right)\right\}<{\lambda }_2\left(p\right)/{\mu }_2\left(p\right)$ for every point p, where μ_i (p) and λ_i (p) are pointwise bounds on rates of expansion and contraction. This pointwise condition always holds when dim E^c = 1, and they show in [37] that this assumption suffices to get the following.

Theorem 8.3

AC² volume-preserving partially hyperbolic essentially accessible (dynamically coherent)center-bunched diffeomorphism is ergodic.

Grayson, Pugh and Shub [49] proved this theorem for small perturbations of the time one map of the geodesic flow on a surface of constant negative curvature. Wilkinson in her thesis extended their result to small perturbations of the time-1 map of the geodesic flow on an arbitrary surface of negative curvature. Then Pugh and Shub in [77, 78] proved the theorem assuming a stronger center bunching condition. The proof of the theorem in the form stated here (with a weaker center bunching condition) was obtained by Burns and Wilkinson in [37].

Remark 8.4

Burns and Wilkinson recently announced that the assumption of dynamical coherence is not needed in Theorem 8.3 [38].

Together with the comments on Definition 8.2 this in particular gives the following.

Corollary 8.5

A C ² volume-preserving essentially accessible partially hyperbolic diffeomorphism with dim E^c = 1 is ergodic.

This corollary was also announced recently by F. Rodriguez Hertz, J. Rodriguez Hertz and R. Ures [82].

The way to establish (8.2) without absolute continuity of the center-stable and center-unstable foliation is through the use of a collection of special sets at every point x ∈ M called Juliennes, J_n (x). ⁵ We shall describe a construction of these sets which assures that

(J1)

J_n (x) form a basis of the topology.

(J2)

J_n (x) form a basis of the Borel σ-algebra. More precisely, let Z be a Borel set; a point x ∈ Z is said to be Julienne dense if

${}_{n\to +\infty }^{\lim }\frac{v\left(J_n\left(x\right)\bigcap Z\right)}{v\left(J_n\left(x\right)\right)}=1.$

Let D (Z) be the set of all Julienne dense points of Z. Then

$D\left(z\right)=z\left(\mathrm{mod}0\right).$

(J3)

If Z ⊂ sat(ℳ^s )∩sat (ℳ^u ), then $\mathcal{D}$ (Z)∈sat (ℳ^s ∩ ℳ^u ).

Properties (J1)–(J3) imply (8.2).

Note that the collection of balls B (x, 1/n) satisfies requirements (J1) and (J2) but not (J3). Juliennes can be viewed as balls "distorted" by the dynamics in the following sense. Fix an integer n ≥ 0, a point x ∈ M and numbers τ, σ such that 0 <τ <σ < 1. Denote by

$\begin{array}{c}{B}_n^s(x,\tau )=\{y\in W^s(x)|\rho (f^{-k}(x),(f^{-k}(y))\le {\tau }^k\},\\ B_n^u(x,\tau )=\{y\in W^u(x)|\rho (f^k(x),f^k(y))\le {\tau }^k\},\end{array}$

and define the Julienne

$J_n\left(x\right):=\left[J_n^{cs}\left(x\right)\times B_n^u\left(x,\tau \right)\right]\bigcap \left[B_n^s\left(x,\tau \right)\times J_n^{cu}\left(x\right)\right],$

where the local foliation products

$\begin{array}{cc}{J}_n^{cs}\left(x\right)=B_n^s\left(x,\tau \right)\times B^c\left(x,{\sigma }^n\right),& J_n^{cu}\left(x\right)=B_n^u\left(x,\tau \right)\times B^c\left(x,{\sigma }^n\right)\end{array}$

are the center-stable and center-unstable Juliennes, and B^c (x, σⁿ ) is the ball in W^c (x) centered at x of radius σⁿ . One may think of J_n (x) as a substitute for $B_n^s\left(x,\tau \right)\times B^c\left(x,{\sigma }^n\right)\times B_n^u\left(x,\tau \right)$ , which is only well defined if the stable and unstable foliations are jointly integrable.

The proof of (J1)–(J3) is based on the following properties of Juliennes:

(1)

scaling:if k ≥ 0 then ν(J_n (x))/ν(J_{n +k} (x)) is bounded, uniformly in n ∈ ℕ

(2)

engulfing: there is ℓ ≥ 0 such that, for any x, y ∈ M, if J_{n +ℓ} (x) ∩ J_{n +ℓ} (y) ≠ ∅ then J_{n +ℓ}(x) U J_{n +ℓ}(y) ∈ J_n (x);

(3)

quasi-conformality: there is k ≥ 0 such that if x, y ∈ M are connected by an arc on an unstable manifold that has length ≤ 1 then the holonomy map π :V^cs (x) → V^cs (y) generated by the family of local unstable manifolds (see Section 3.2) satisfies $J_{n+k}^{cs}\left(y\right)\subset \pi \left(J_n^{cs}\left(x\right)\right)\subset J_{n-k}^{cs}\left(y\right)$ .

The properties (1) and (2) are possessed by the family of balls in Euclidean space and they underlie the proof of the Lebesgue Density Theorem. One can use these properties to show that Juliennes are density bases. The center-unstable Juliennes are a density basis on W^cu (x) with respect to the smooth conditional measure $v_{W^{cu}}$ on W^cu (x), the center-stable Juliennes are a density basis on W^cs (x) with respect to the smooth conditional measure $v_{W^{cs}}$ on W^cs (x), and the Juliennes are a density basis on M with respect to the smooth measure ν.

Juliennes, J_n (x), are small but highly eccentric sets in the sense that the ratio of their diameter to their inner diameter increases with n (the inner diameter of a set is the diameter of the largest ball it contains). In general, sets of such shape may not form density bases, but Juliennes do because their elongation and eccentricity are controlled by the dynamics; in particular, they nest in a way similar to balls.

Quasi-conformality is what is needed to prove Property (J3). Roughly speaking it means that the holonomy map (almost) preserves the shape of Juliennes.

Conjecture 8.6

A partially hyperbolic dynamical system preserving a smooth measure and with the accessibility property is ergodic.

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Time Series Analysis: Methods and Applications

Suhasini Subba Rao , in Handbook of Statistics, 2012

2.2.2 Locally stationary time series

In this section, we show that a subclass of SCR models and the class of locally stationary linear processes defined by Dahlhaus (1996) are closely related. We restrict the regressors to be smooth, and assume there exists functions $\mathfrak{\{}{x}_j\left(\cdot \right)\mathfrak{\}}$ such that the regressors satisfy $x_{t,j}=x_j\left(\frac{t}{N}\right)$ for some value N (setting $\frac{1}{T}{\sum }_t{x}_{t,j}^2=1$ ) and $\mathfrak{\{}{Y}_{t,N}\mathfrak{\}}$ satisfies

(4) $Y_{t,N}=\sum _{j=1}^n{a}_j{x}_j\left(\frac{t}{N}\right)+X_{t,N},\text{\hspace{0.17em}where}{X}_{t,N}=\sum _{j=1}^n{\alpha }_{t,j}{x}_j\left(\frac{t}{N}\right)+{\epsilon }_{t}t=1,\dots ,T.$

A nonstationary process can be considered locally stationary process, if in any neighborhood of t, the process can be approximated by a stationary process. We now show that $\mathfrak{\{}{X}_{t,N}\mathfrak{\}}$ (defined in (4)) can be considered as a locally stationary process.

Proposition 1

Suppose Assumption 1 (i,ii) is satisfied, and the regressors are bounded ( $\underset{j,v}{\sup }\mathfrak{|}{x}_j\left(v\right)\mathfrak{|}<\mathfrak{\infty }$ ), let $X_{t,N}$ be defined as in (4) and define the unobserved stationary process $X_t\left(v\right)={\sum }_{j=1}^n{\alpha }_{t,j}{x}_j\left(v\right)+{\epsilon }_t$ . Then we have

$\mathfrak{|}{X}_{t,N}-X_t\left(v\right)\mathfrak{|}=O_p\left(\left|\right.\frac{t}{N}-v\left|\right.\right).$

Proof

Using the Lipschitz continuity of the regressors the proof is straightforward, hence we omit the details.

The above result shows that in the neighborhood of t, $\mathfrak{\{}{X}_{t,N}\mathfrak{\}}$ can locally be approximated by a stationary process. Therefore the SCR model with slowly varying regressors can be considered as a "locally stationary" process.

We now show the converse, that is the class of locally stationary linear processes defined by Dahlhaus (1996), can be approximated to any order by an SCR model with slowly varying regressors. Dahlhaus (1996) defines locally stationary process as the stochastic process $\mathfrak{\{}{X}_{t,N}\mathfrak{\}}$ , which satisfies the representation

(5) $X_{t,N}=\int A_{t,N}\left(\omega \right)\exp \left(it\omega \right)dZ\left(\omega \right),$

where $\mathfrak{\{}Z\left(\omega \right)\mathfrak{\}}$ is a complex-valued orthogonal process on $\left[0,2\pi \right]$ with $Z\left(\lambda +\pi \right)=Z\left(\lambda \right)$ , $\mathbb{E}\left(Z\left(\lambda \right)\right)=0$ , and $\mathbb{E}\mathfrak{\{}dZ\left(\lambda \right)dZ\left(\mu \right)\mathfrak{\}}=\eta \left(\lambda +\nu \right)d\lambda d\mu$ , $\eta \left(\lambda \right)={\sum }_{j=-\mathfrak{\infty }}^{\mathfrak{\infty }}\delta \left(\lambda +2\pi j\right)$ is the periodic extension of the Dirac delta function. Furthermore, there exists a Lipschitz continuous function $A\left(\cdot \right)$ , such that $\underset{\omega ,t}{\sup }\mathfrak{|}A\left(\frac{t}{N},\omega \right)-A_{t,N}\left(\omega \right)\mathfrak{|}\le K{N}^{-1}$ , where K is a finite constant that does not depend on N.

In the following lemma we show that there always exists a SCR model that can approximate a locally stationary process to any degree.

Proposition 2

Let us suppose that $\mathfrak{\{}{X}_{t,N}\mathfrak{\}}$ is a locally stationary process that satisfies (5) and $\underset{u}{\sup }\int \mathfrak{|}A\left(u,\lambda \right){\mathfrak{|}}^{2}d\lambda <\mathfrak{\infty }$ . Then for any basis $\mathfrak{\{}{x}_j\left(\cdot \right)\mathfrak{\}}$ of $L_2\left[0,1\right]$ , and for every δ there exists an $n_{\delta }\mathfrak{\in }\mathbb{Z}$ , such that $X_{t,N}$ can be represented as

(6) $X_{t,N}=\sum _{j=1}^{n_{\delta }}{\alpha }_{t,j}{x}_j\left(\frac{t}{N}\right)+O_p(\delta +N^{-1}),$

where $\mathfrak{\{}{\alpha }_t\mathfrak{\}}=\mathfrak{\{}({\alpha }_1,\dots ,{\alpha }_{t,n_{\delta }}){\mathfrak{\}}}_t$ is a second-order stationary vector process.

Proof

In the technical report.

One application of the above result is that if the covariance structure of a time series is believed to change smoothly over time, then a SCR model can be fitted to the observations.

In the sections below, we will propose a method of estimating the parameters in the SCR model and use the SCR model with slowly varying parameters as a means of comparing the proposed method with existing Gaussian likelihood methods.

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Locally Stationary Processes and the Local Block Bootstrap

Arif Dowla , ... Dimitris N. Politis , in Recent Advances and Trends in Nonparametric Statistics, 2003

2 CONSISTENCY OF THE LOCAL BLOCK BOOTSTRAP

As is well-known, the applicability of bootstrap methods is usually checked on a case-by-case basis—the only exception so far seems to be the i.i.d. bootstrap with smaller resample size that is generally consistent; see e.g. Politis, Romano and Wolf (1999). Thus, we now focus on a particular interesting example of a locally stationary process in the sense of Dahlhaus (1996, 1997). Let, the data be generated from the model ²

(1) $Y_t=s_n\left(t\right)+\mu +v_n\left(t\right)e_t,\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{all}t\in \mathbf{Z}$

where {e_t , t ∈ Z} is a mean-zero and variance-one, strong mixing and strictly stationary sequence satisfying:

$E\left|e_t\right|{}^{6+\delta }<\infty ,\mathrm{and}{\displaystyle \sum _{k=1}^{\infty }{k}^2{\alpha }^{\delta /\left(6+\delta \right)}}\left(k\right)<\infty \mathrm{f}\mathrm{o}\mathrm{r}\mathrm{some}\delta >0\text{,}$

where α(k) indicates the strong mixing coefficient associated with {e_t , t ∈ Z}. We also assume that s_n (t)   = m(t/n), v_n (t)   = σ(t/n), for some fixed functions m, σ that, are differentiable ³ with bounded derivatives on [0,1], We will further assume that σ(x) >   0 for all x in [0,1] and that m satisfies ${\int }_0^{1}m\left(x\right)dx=0$ ; due the latter, m may be thought of as a 'seasonality' fluctuation about the 'grand mean' μ.

Our goal is interval estimation of the unknown parameter μ based on the data Y ₁, …., Y_n . For this reason we require an approximation to the sampling distribution of the sample mean $\widehat{\mu }=n^{-1}{\displaystyle {\sum }_{t=1}^n{Y}_t}$ that will serve as our estimator of μ. We propose the LBB as a method for this approximation.

We first establish some of the properties of $\widehat{\mu }$ . The following two lemmas concern the asymptotic bias and variance of the sample mean.

Lemma 1.

$E\left(n^{\frac{1}{2}}\left(\widehat{\mu }-\mu \right)\right)=O\left(n^{-\frac{1}{2}}\right)\text{.}$

Lemma 2.

Let c(s)   = Cov(e ₀, e_s ). Then, as n  →   ∞, $Var\left(n^{\frac{1}{2}}\left(\widehat{\mu }-\mu \right)\right)\to V^2$ , where $V^2={\displaystyle \sum _{s=-\infty }^{\infty }c\left(s\right){\displaystyle {\int }_0^1{\sigma }^2\left(u\right)du.}}$

In the next theorem we establish asymptotic normality of the sample mean.

Theorem 1

As n  →   ∞ $\sqrt{n}\left(\widehat{\mu }-\mu \right)\stackrel{ℒ}{\Rightarrow }N\left(0,V^2\right)$

We now turn to the LBB properties. Let Y*₁, …, Y_n * denote an LBB pseudo-series constructed using the algorithm of the previous section, and let $\widehat{\mu }*=n^{-1}{\displaystyle {\sum }_{t=1}^n{Y}_t^{*}}$ be the bootstrap sample mean. As usual, let P*, E*, Var* indicate probability, expectation, and variance under the LBB scheme (conditional on the data Y ₁, …, Y_n ).

To investigate the consistency of the LBB we will—as previously mentioned—require some conditions on the block size, as well as the window size indicating the local neighborhood. For this reason, consider the following:

(2) $\text{Let}n\to \infty ,b\to \infty \text{but}b=0\left(\min (n^{\frac{2}{5}},nB)\right)\to 0,\text{and}nB\to \infty \text{but}n{B}^2\to 0.$

Lemma 3 below can be compared to Lemma 1 and 2.

Lemma 3.

Under (2), $E^{*}\left(n^{\frac{1}{2}}\left({\widehat{\mu }}^{*}-\mu \right)\right)=O_p\left(n{B}^2\right)$ , and $Va{r}^{*}\left(n^{\frac{1}{2}}\left({\widehat{\mu }}^{*}-\widehat{\mu }\right)\right)\stackrel{p}{\to }{V}^2$ .

Our main theorem below shows that the LBB is successful in giving a consistent approximation to the sampling distribution of the sample mean $\widehat{\mu }$ .

Theorem 2

Under (2), we have

(3) $\underset{x}{sup}|P\sqrt{n}\left(\widehat{\mu }-\mu \right)\le x)-P*\left(\sqrt{n}\left(\widehat{\mu }*-E*\widehat{\mu }*\right)\le x\right)|\stackrel{P}{\to }0\text{,}$

as well as

(4) $\underset{x}{sup}|P\sqrt{n}\left(\widehat{\mu }-\mu \right)\le x)-P*\left(\sqrt{n}\left(\widehat{\mu }*-\widehat{\mu }\right)\le x\right)|\stackrel{P}{\to }0\text{.}$

From Theorem 2 it, follows that, asymptotically valid confidence intervals for μ can be based on the quantiles of either the bootstrap distribution $P^{*}\left(\sqrt{n}\left({\widehat{\mu }}^{*}-E^{*}{\widehat{\mu }}^{*}\right)\le x\right)$ or the bootstrap distribution $P^{*}\left(\sqrt{n}\left({\widehat{\mu }}^{*}-\widehat{\mu }\right)\le x\right)$ , both of which are computable—as opposed to the quantiles of the unknown true distribution $P^{*}\left(\sqrt{n}\left(\widehat{\mu }-\mu \right)\le x\right)$ .

To help delineate which of those two approximations may be preferrable, recall that, the distribution $P^{*}\left(\sqrt{n}\left({\widehat{\mu }}^{*}-\widehat{\mu }\right)\le x\right)$ has center of location O_p (nB ²) by Lemma 3, while $P^{*}\left(\sqrt{n}\left({\widehat{\mu }}^{*}-E^{*}{\widehat{\mu }}^{*}\right)\le x\right)$ has center exactly zero, and $P\left(\sqrt{n}\left(\widehat{\mu }-\mu \right)\le x\right)$ has center $O\left(1/\sqrt{n}\right)$ by Lemma 1. In order to satisfy (2) we may let B =   1/n ^{ϵ  +   1/2} for some ϵ in (0,1/2). Typically, we may even have ϵ <   1/4, in which case approximation (3) is preferrable as its (zero) center of location is closest, to that of the target, distribution $P^{*}\left(\sqrt{n}\left(\widehat{\mu }-\mu \right)\le x\right)$ . In addition, the validity of (3) may be proved under slightly weaker conditions, namely replacing the condition nB ²  →   0 in (2) by the weaker B  →   0.

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Risk Model with Variable Premium Intensity and Investments in One Risky Asset

Yuliya Mishura , Olena Ragulina , in Ruin Probabilities, 2016

7.3 Existence and uniqueness theorem

Consider now equation [7.3]. Let t*(x) be a possible explosion time of (X_t (x)) _{t  ≥   0}, i.e.

$t^{*}\left(x\right)=inf\left\{t\ge 0:X_t\left(x\right)\notin \left(-\infty ,+\infty \right)\right\}\text{.}$

To shorten notation, we let t* stand for t*(x).

Theorem 7.2

If c(u) is a locally Lipschitz continuous function on ℝ, then [7.3] has a unique strong solution up to the time τ ∧ t*.

Proof

Since the process (N_t ) _{t  ≥   0} is homogeneous, it has only a finite number of jumps on any finite time interval a.s. To prove the theorem, we study [7.3] between two successive jumps of (N_t ) _{t  ≥   0}.

Let us first consider [7.3] on the time interval [τ ₀ , τ ₁), where τ ₀  =   0. It can be rewritten as

[7.16] $X_t=X_{{\tau }_0}+{\int }_{{\tau }_0}^t\left(c\left(X_s\right)+a{X}_s\right)\mathrm{d}s+b{\int }_{{\tau }_0}^t{X}_s\mathrm{d}{W}_s,{\tau }_0\le t\le {\tau }_1\text{.}$

By theorem A.8, the locally Lipschitz continuity of c(u)   + au and bu on ℝ implies the existence of a unique strong solution to [7.16] on [τ ₀, τ ₁ ∧ t*). Moreover, the comparison theorem (see section A.3.2) shows that this solution is not less than the solution to the equation

[7.17] $X_t=X_{{\tau }_0}+a{\int }_{{\tau }_0}^t{X}_s\mathrm{d}s+b{\int }_{{\tau }_0}^t{X}_s\mathrm{d}{W}_s,{\tau }_0\le t\le {\tau }_1\wedge t^{*}\text{,}$

a.s. Since the solution to [7.17] is positive, so is the solution to [7.16] on [τ ₀, τ ₁ ∧ t*). Hence, lim _t↑t* X_t   =   +∞ if t*   ≤ τ ₁. Thus, ruin does not occur up to the time τ ₁ ∧ t*.

If t*   ≤ τ ₁, then the theorem follows. Otherwise $X_{{\tau }_{1-}}<+\infty$ and we set $X_{{\tau }_1}=X_{{\tau }_{1-}}-Y_1$ . Next, if X _τ1  <   0, then τ  = τ ₁, which completes the proof. Otherwise we consider [7.3] on the time interval [τ ₁, τ ₂). We rewrite it as

[7.18] $X_t=X_{{\tau }_1}+{\int }_{{\tau }_1}^t\left(c\left(X_s\right)+a{X}_s\right)\mathrm{d}s+b{\int }_{{\tau }_1}^t{X}_s\mathrm{d}{W}_s,{\tau }_1\le t<{\tau }_2\text{.}$

Repeating the same arguments, we conclude that [7.18] has a unique strong solution on [τ ₁, τ ₂ ∧ t*) and ruin does not occur up to the time τ ₂ ∧ t*.

Thus, we have proved that [7.3] has a unique strong solution on [0, τ ₂ ∧ t*), which is our assertion if t*   ≤ τ ₂. For the case t*   > τ ₂, we set $X_{{\tau }_2}=X_{{\tau }_{2-}}-Y_2$ . Next, if $X_{{\tau }_2}<0$ , then τ  = τ ₂, which proves the theorem. Otherwise we continue in this fashion and prove the theorem by induction.

Remark 7.4

Note that if t*   <   ∞, then the proof of theorem 7.2 implies lim _t↑t* X_t   =   +∞ and [7.3] also holds for t  = t* provided that we let both of its sides be formally equal to   +   ∞. In this case, we formally set X _t*  =   +∞. In addition, if τ  <   ∞, then we set $X_{\tau }=X_{{\tau }_{i-}}-Y_i$ , where i is the number of the claim that caused the ruin, and [7.3] also holds for t  = τ.

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Conditional density estimation

Dag Tjøstheim , ... Bård Støve , in Statistical Modeling Using Local Gaussian Approximation, 2022

10.5 Proof of theorems

10.5.1 Proof of Theorem 10.1

Except a slight modification that accounts for the replacement of independence with α-mixing, the proof of Theorem 10.1 is identical to the corresponding proof of Theorem 9.1, which again is based on the global maximum likelihood case covered by Severini (2000). For each location z , which we suppress from notation for simplicity, denote by $\mathrm{\text{E}}(L_{{\mathit{b}}_n,K}(\rho ))$ the expectation of the local likelihood function $L_n(\rho ,\mathit{Z})$ . The consistency follows from the uniform convergence in probability of $L_n(\rho ,\mathit{Z})$ toward $\mathrm{\text{E}}(L{\mathit{b}}_n,K(\rho ))$ , conditions for which are provided in Corollary 2.2 by Newey (1991).

The result requires a compact support of the parameter space, the equicontinuity, and Lipschitz continuity of the family of functions $\{\mathrm{\text{E}}(L_{{\mathit{b}}_n,K}(\rho ))\}$ as well as the pointwise convergence of the local likelihood functions. The compactness is covered by assumption (iv), and the demonstration of equicontinuity and Lipschitz continuity in the proof of Theorem 9.1 does not rely on the independent data assumption. The pointwise convergence follows from a standard non-parametric law of large numbers in the independent case. However, our assumption (ii) about α-mixing data ensures that pointwise convergence still holds; see, for example, Theorem 1 by Irle (1997), conditions for which are straightforward to verify in our local likelihood setting.   □

10.5.2 Proof of Theorem 10.2

Consider first the bivariate case, in which there is only one local correlation to estimate. The first part of the proof goes through exactly as in the iid case in Theorem 9.2 of Chapter 9. We follow the argument for global maximum likelihood estimators as presented in Theorem 7.63 by Schervish (1995). The statement of Theorem 10.2 follows provided that

(10.9) $Y_n(\mathit{z})=\sum _{t=1}^n{K}_{{\mathit{b}}_n}({\mathit{z}}_t-\mathit{z})u({\mathit{z}}_t,{\rho }_0)\stackrel{\mathrm{\text{def}}}{=}\sum _{t=1}^n{V}_{nt}$

is asymptotically normal, and this follows from a standard Taylor expansion. In the iid case, the limiting distribution of (10.9) is derived using the same technique as when demonstrating the asymptotic normality for the standard kernel estimator, for example, as in the proof of Theorem 1A by Parzen (1962). We establish the asymptotic normality of (10.9) in the case of α-mixing data by going through the steps used in proving Theorem 2.22 in Fan and Yao (2003). Let $W_t=b_n^{-1}{V}_{nt}$ , where $b_n$ is the component bandwidth in ${\mathit{B}}_n=\mathrm{\text{diag}}({\mathit{b}}_n)$ . Then

$\frac{1}{n{b}_n^2}\mathrm{\text{Var}}(Y_n(\mathit{z}))=\frac{1}{n{b}_n^2}\{\sum _{t=1}^n\mathrm{\text{Var}}(V_{nt})+2\sum {\sum }_{1⩽t<s⩽n}\mathrm{\text{Cov}}(V_{nt},V_{ns})\}=\mathrm{\text{Var}}(W_1)+2\sum _{s=1}^n(1-s/n)\mathrm{\text{Cov}}(W_1,W_{s+1}),$

where

$\mathrm{\text{Var}}(W_1)=\mathrm{\text{E}}(W_1^2)-{(\mathrm{\text{E}}(W_1))}^2=\int b_n^{-2}{u}^2(\mathit{z},{\rho }_0)K^2(b_n^{-1}(\mathit{y}-\mathit{z}))f(y)\mathrm{\text{d}}\mathit{y}+O(b_n^2)=\int u^2(\mathit{z}+b_n\mathit{v},{\rho }_0)K^2(\mathit{v})f(\mathit{z}+b_n\mathit{v})\mathrm{\text{d}}\mathit{v}+O(b_n^2)\to u^2(\mathit{z},{\rho }_0)f(\mathit{z})\int K^2(\mathit{v})\mathrm{\text{d}}\mathit{v}\stackrel{\mathrm{\text{def}}}{=}M(\mathit{z})\mathrm{\text{as}}{b}_n\to 0,$

and

$|\mathrm{\text{Cov}}(W_1,W_{s+1})|=|\mathrm{\text{E}}(W_1{W}_{s+1})-\mathrm{\text{E}}(W_1)\mathrm{\text{E}}(W_{s+1})|=O(b_n^2)$

using the same argument. Therefore

$|\sum _{s=1}^{m_n}\mathrm{\text{Cov}}(W_1,W_{s+1})|=O(m_n{b}_n^2).$

Fan and Yao (2003) require that

(10.10) $\mathrm{\text{E}}(u{({\mathit{Z}}_t,{\rho }_0(\mathit{z}))}^{\delta })<\infty$

for some $\delta >2$ , and this is of course true for our transformed data, because it is marginally normal. In Proposition 2.5(i) by Fan and Yao (2003), we can therefore use $p=q=\delta >2$ to obtain, for some constant C,

$|\mathrm{\text{Cov}}(W_1,W_{s+1})|⩽C\alpha {(s)}^{1-2/\delta }{b}_n^{4/\delta -2}.$

Let $m_n={(b_n^2|\log b_n^2|)}^{-1}$ . Then $m_n\to \infty$ , $m_n{b}_n^2\to 0$ , and

$\sum _{s=m_n+1}^{n-1}|\mathrm{\text{Cov}}(W_1,W_{s+1})|⩽C\frac{b_n^{4/\delta -2}}{m_n^{\lambda }}\sum _{s=m_n+1}^n{s}^{\lambda }\alpha {(s)}^{1-2/\delta }\to 0,$

which follows from assumption (ii). Thus

$\sum _{s=1}^{n-1}\mathrm{\text{Cov}}(W_1,W_{s+1})\to 0,$

and it follows that

$\frac{1}{n{b}_n^2}\mathrm{\text{Var}}(Y_n(\mathit{z}))=M(\mathit{z})(1+o(1)).$

The proof now continues exactly as in Fan and Yao (2003) using the "big block–small block" technique, but with the obvious replacement of $b_n$ by $b_n^2$ to accommodate the bivariate case.

We expand the argument to the multivariate case using the Cramér–Wold device. Let $\mathit{\rho }={({\rho }_1,\dots ,{\rho }_m)}^T$ be the vector of local correlations, where $m=p(p-1)/2$ , write $\mathit{u}(\mathit{z},{\mathit{\rho }}_0)=(u_1(\mathit{z},{\mathit{\rho }}_0),\dots ,u_m(\mathit{z},{\mathit{\rho }}_0))$ , and let ${\mathit{S}}_n(\mathit{z})={\{S_{ni}(z)\}}_{i=1}^m$ , where

$S_{ni}=\sum _{t=1}^n{u}_i({\mathit{Z}}_t,{\mathit{\rho }}_{0,i})K_{{\mathit{b}}_n}({\mathit{Z}}_t-\mathit{z}).$

We must show that

(10.11) $\sum _{k=1}^m{a}_k{S}_{nk}\stackrel{d}{\to }\sum _{k=1}^m{a}_k{Z}_k^{⁎},$

where $\mathit{a}={(a_1,\dots ,a_m)}^T$ is an arbitrary vector of constants, and ${\mathit{Z}}^{⁎}=(Z_1^{⁎},\dots ,Z_m^{⁎})$ is a jointly normally distributed random vector. It suffices to show that the left-hand side of (10.11) is asymptotically normal, and this follows from observing that it is of the same form as the original sequence comprising $S_n$ with

$\sum _k{a}_k{S}_{nk}=\sum _t{\mathit{u}}^{⁎}(Z_t,{\mathit{\rho }}_0)K_{{\mathit{b}}_n}({\mathit{Z}}_t-\mathit{z}),$

where ${\mathit{u}}^{⁎}({\mathit{Z}}_t,{\mathit{\rho }}_0)={\sum }_k{a}_k{u}_k({\mathit{Z}}_t,{\mathit{\rho }}_0)$ . It is well known that any measurable mapping of a mixing sequence of random variables inherit the mixing properties of the original series, so condition (ii) is therefore satisfied by the linear combination. The new sequence of observations satisfies (10.10) because it follows from Jensen's inequality that for any even integer $\delta >2$ (making the function $f(x)=x^{\delta }$ convex),

${\left[\frac{{\mathit{u}}^{⁎}({\mathit{Z}}_t,{\mathit{\rho }}_0)}{{\sum }_k{a}_k}\right]}^{\delta }={\left[\frac{{\sum }_k{a}_k{u}_k({\mathit{Z}}_t,{\mathit{\rho }}_0)}{{\sum }_k{a}_k}\right]}^{\delta }⩽\frac{{\sum }_k{a}_k{[u_k({\mathit{Z}}_t,{\mathit{\rho }}_0)]}^{\delta }}{{\sum }_k{a}_k},$

so that

$\mathrm{\text{E}}{[u^{⁎}({\mathit{Z}}_t,{\mathit{\rho }}_0)]}^{\delta }⩽\sum _k{a}_k\mathrm{\text{E}}{[u_k({\mathit{Z}}_t,{\mathit{\rho }}_0)]}^{\delta }{\left[\sum _k{a}_k\right]}^{\delta -1}<\infty .$

Hence condition (10.10) is fulfilled, and the results follow from Theorem 2.22 in Fan and Yao (2003).

The off-diagonal elements in the asymptotic covariance matrix are asymptotically zero by the same arguments as in the proof of Theorem 9.3 in Chapter 9.   □

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Navier–Stokes Equations

In Studies in Mathematics and Its Applications, 1977

Imbedding Theorems

We recall the Sobolev imbedding theorems which will be used frequently from now on. Let m be an integer and p any finite number greater than or equal to one, p ⩾ 1; then, if 1/p - m/n = 1/q > 0 the space W ^{m, p} ( $ℛ$ ⁿ ) is included in L^q ( $ℛ$ ⁿ ) and the injection is continuous. If u ∈ W ^{m, p} ( $ℛ$ ⁿ ) and 1/p - m/n = 0 then u belongs to L^q ( $O$ ) for any bounded set $O$ and any q, 1 ≤ q < ∞. If 1/p - m/n < 0 then a function in W ^{m, p} ( $ℛ$ ⁿ ) is almost everywhere equal to a continuous function; such a function has also some Hölder or Lipschitz continuity properties but these properties will not be used here; if a function belongs to W ^{m, p} ( $ℛ$ ⁿ ) with 1/p - m/n < 0 then the derivatives of order α belong to W ^m-α,p ( $ℛ$ ⁿ ) and some embedding results of preceding type hold for these derivatives if 1/p – (m – α)/n > 0.

For u ∈ W ^{m, p} ( $ℛ$ ⁿ ), m ⩾ 1, 1 ≤ p < ∞

(1.1) $\begin{array}{l}if\frac{1}{p}-\frac{m}{n}=\frac{1}{q}<0,{\left|u\right|}_{L^q\left({\Re }^n\right)}⩽c\left(m,p,n\right){║u║}_{w^{m,p\left({\Re }^n\right)}},\\ if\frac{1}{p}-\frac{m}{n}=0,{\left|u\right|}_{L^q\left(0\right)}⩽c\left(m,p,n,q,o\right){║u║}_{w^{m,p\left({\Re }^n\right)}}\\ \forall boundedsetO\subset {\Re }^n,{\forall }_q,1⩽q⩽\infty ,\\ if\frac{1}{p}-\frac{m}{n}<0,{\left|u\right|}_{C^{\circ }\left(O\right)}⩽c\left(m,n,p,O\right){║u║}_{w^{m,p\left({\Re }^n\right)}},\\ \forall boundedsetO,O\subset {\Re }^n.\end{array}$

If Ω is any open set of $ℛ$ ⁿ , results similar to (1.1) can usually be obtained if Ω is sufficiently smooth so that:

(1.2) $\begin{array}{l}Thereexistsacontinuouslinearprolongationoperator\\ \Pi \in L\left(W^{m,p}(\Omega ),W^{m,p}\left({\Re }^n\right)\right).\end{array}$

Property (1.2) is satisfied by a locally Lipschitz set Ω. When (1.2) is satisfied, the properties (1.1) applied to Πu, u ∈ W ^{m, p} (Ω) give in particular, assuming that u ∈ W ^{m, p} (Ω), m ⩾ 1, 1 < p < ∞, and (1.2) holds:

(1.3) $\begin{array}{l}if\frac{1}{p}-\frac{m}{n}=\frac{1}{q}<0,{\left|u\right|}_{L^q(\Omega )}⩽c\left(m,p,n,\Omega \right){║u║}_{w^{m,p(\Omega )}},\\ if\frac{1}{p}-\frac{m}{n}=0,{\left|u\right|}_{L^q\left(O\right)}⩽c\left(m,p,n,q,O,\Omega \right){║u║}_{w^{m,p(\Omega )}},\\ anyq,1⩽q<\infty ,anyboundedsetO\subset \overline{\Omega },\\ f\frac{1}{p}-\frac{m}{n}<0,{\left|u\right|}_{C^{\circ }\left(O\right)}⩽c\left(m,p,n,q,\Omega ,O\right){║u║}_{W^{m,p(\Omega )}},\\ anyboundedsetO,O\subset \overline{\Omega }.\end{array}$

When $u\in \stackrel{\circ }{{\stackrel{\circ }{W}}^{m,p}(\Omega )},$ , the function ũ which is equal to u in Ω and to 0 in

Ω, belongs to W ^{m, p} ( $ℛ$ ⁿ ), and hence the properties (1.3) are valid without any hypothesis on Ω.

The case of particular interest for us is the case p = 2, m = 1, i.e., the case H ₀ ¹(Ω). Without any regularity property required for Ω we have for u ∈ H ₀ ¹(Ω)

(1.4) $\begin{array}{l}n=2,{\left|u\right|}_{L^q(\Omega )}⩽c\left(q,O,\Omega \right){║u║}_{H_0^1(\Omega )}\\ \forall boundedsetO\subset \Omega ,\forall q,1⩽q⩽\infty \\ n=3,{\left|u\right|}_{L^6(\Omega )}⩽c(\Omega ){║u║}_{H_0^1(\Omega )}\\ n=4,{\left|u\right|}_{L^4(\Omega )}⩽c\left(\Omega \right){║u║}_{H_0^1(\Omega )}\\ n⩾3,\left|u\right|L^{2n/\left(n-2\right)}(\Omega )⩽c(\Omega ){║u║}_{H_0^1(\Omega )}.\end{array}$

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Handbook of Mathematical Fluid Dynamics

Denis Serre , À ma Mère , in Handbook of Mathematical Fluid Dynamics, 2007

Vertex at the origin.

The boundary of the physical domain has a singularity at the origin. It will be responsible for a lack of smoothness of the solution. In view of Equation (118), it is unclear whether the pressure has a singularity, or the velocity vanishes.

Sonic line.

This difficulty arises in a supersonic RR. Across the sonic line S, the type of the Euler system changes from hyperbolic (in the supersonic zone labelled 2) to hyperbolic-elliptic (in the subsonic domain D). In D, the elliptic part of the system degenerates as one approaches the boundary. When the strength of the incident shock is small, we are tempted to use linear or weakly non-linear geometrical optics (see next Paragraph), which suggest that the flow is Hölder continuous across the sonic line, with a singularity of order $\sqrt{d\left(y;S\right)}$ where d(·; S) is the distance to S in D; see for instance the acoustic approximation of J. Keller and A. Blank [49]. This picture turns out to be incorrect. It has been uncovered recently by G.-Q. Chen and M. Feldman [22 ] that the nonlinearity in the system yields Lipschitz continuity, at least in the potential case, where the system reduces to a nonlinear wave equation.

Vertex at P.

Alternatively, in a transonic RR, the subsonic zone extends till P, and we have to solve a nonlinear system of PDEs in a domain that has a vertex at P.

Mixed type.

In the subsonic domain D, the system is of mixed type hyperbolic-elliptic, meaning that one characteristic field is real, while the other ones are complex. This makes the analysis very hard, since it is not possible either to apply standard techniques of hyperbolic problems or to employ ideas from elliptic theory.

Vortical singularity.

We shall see in Section 6.6 that the vorticity ∇ × u cannot be square integrable in D, at least in the barotropic case. A singularity is expected at some point along the ramp. It is unclear whether this singularity is present in full gas dynamics.

Slip line.

In Mach Reflection, the interaction point P (defined as the point where the reflected and the incident shocks meet) is detached, and a secondary shock called Mach stem connects P to the ramp. Since the Rankine–Hugoniot relations are the same as in the steady case, a pure three-shocks pattern is not possible (Theorem 2.3). Hence a fourth wave must originate from P. Experiments suggest that it is a slip line, or in other words a vortex sheet. According to M. Artola and Majda [2], such jumps are known to be dynamically unstable unless the jump of the tangential velocity exceeds $2\sqrt{2}c$ , which is unlikely ¹⁸ . Finally, one observes that the slip line rolls up endlessly.

Diffracted shock.

The solution is known everywhere but in the subsonic zone D. However, the part of the boundary of D formed by the diffracted shock is a free boundary. For an incident shock of small strength, this curve is approximately a circle, the continuation of the sonic line.

Triple point.

The sonic line and the reflected shock form a corner at their meeting point Q. This is a singularity of the boundary of the subsonic domain. This singularity is non uniform in terms of the shock strength, as both lines tend to become tangent when the strength vanishes.

We point out that the vortical singularity and the slip line are obviously not present in irrotational flows. Additionally, the system becomes purely elliptic in D. This makes the irrotational RR much more tractable, with only the difficulties of non-uniform ellipticity, boundary vertex, triple point and free boundary. In particular, one may expect that the flow be of class H ¹ within D. Zheng [86], as well as Chen and Feldman [22] obtained recently an existence result in this case, when the incident shock is almost normal.

Another simplification occurs in the transonic case (|u ₂ − P| < c ₂). Then we avoid the degeneracy problem across the sonic line and the triple point Q. In conclusion, the simplest situation for a RR is that of an irrotational flow for which the subsonic zone reaches the point P. Then the mathematical problem is to solve a scalar second order nonlinear elliptic equation in terms of the potential. The remaining difficulties are the free boundary and two geometrical singularities, at O and P. We point out on the one hand that this problem does not seem to follow from the minimisation of some action. On the other hand, it happens in some intermediate (very narrow) range of parameters and thus is not a perturbation of some trivial configuration. Therefore it cannot be attacked by pertubative tools.

Finally, let us mention the work by S. Chen [24], who proves the existence of a local solution for the reflexion of a shock against a smooth convex obstacle. Of course, this result is sensitive to the curvature of the boundary, and does not survive when the shape of the obstacle becomes sharp. S. Chen also proved a local stability result, near the triple point, of a Mach configuration; this result is in the spirit of Paragraph 3.3.

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Stationary Partial Differential Equations

Patrizia Pucci , James Serrin , in Handbook of Differential Equations: Stationary Partial Differential Equations, 2007

Theorem 3.5.1 (Comparison Principle)

Let Q = ∅ or Q = {0}. Suppose that A = A (x, ξ) is independent of z and strictly elliptic in Ω × R _r for all r > 0. Assume additionally that B(x, z, ξ) is locally Lipschitz continuous with respect to ξ in Ω × ℝ × ℝ ⁿ and moreover is non-increasing in z.

Let u and v be solutions of (3.5.1) and (3.5.2) of class $W_{\text{loc}}^{1,\infty }\left(\Omega \right)$ in Ω. If u ⩽ v + M on ∂Ω, where M is constant, then u ⩽ v + M in Ω.

Proof

We treat only the case Q = {0}. When Q is empty the proof is slightly simpler, and can be omitted. moreover, since A is independent of z it is enough to consider only the case M = 0.

Step 1. Suppose that (x, ξ′), (x, η′) ∈ K′ = Ω × R _W for some W > 0, [ξ′, η′], given by

(3.5.4) $\begin{array}{cc}\zeta \left(t\right)=t{\xi }^{\prime }+\left(1-t\right){\eta }^{\prime },& t\in \left[0,1\right],\end{array}$

does not include 0, then by the mean value theorem

$〈A\left(x,{\xi }^{\prime }\right)-A\left(x,{\eta }^{\prime }\right),{\xi }^{\prime }-{\eta }^{\prime }〉=〈{\partial }_{\xi }A\left(x,\zeta \right)\left({\xi }^{\prime }-{\eta }^{\prime }\right),{\xi }^{\prime }-{\eta }^{\prime }〉$

for some ζ ∈ [ξ′, η′]. Since by hypothesis the matrix [∂_ξ A(x, ξ)] is uniformly positive definite in Ω × R _W , it follows that

(3.5.5) $〈A\left(x,{\xi }^{\prime }\right)-A\left(x,{\eta }^{\prime }\right),{\xi }^{\prime }-{\eta }^{\prime }〉⩾a_1{\left|{\xi }^{\prime }-{\eta }^{\prime }\right|}^2,$

where

$a_1=\underset{x\in \Omega ,\xi \in R_W}{\inf }\left\{\text{min\hspace{0.17em}eigenvalue\hspace{0.17em}of}\left[{\partial }_{\xi }A\left(x,\xi \right)\right]\right\}>0.$

We claim that (3.5.5) holds also when 0 ∈ [ξ′, η′]. First, if 0 is an end point of [ξ′, η′], say η′ = 0, it is enough to let η′ → 0 in (3.5.5), since A is continuous at 0 and a ₁ remains unchanged. The remaining possibility, when 0 is in the interior of [ξ′, η′] is now obvious.

Next, if (x, u, ξ′), (x, v, η′) ∈ K″, where K″ is a compact subset of Ω × ℝ × ℝ ⁿ , then by local Lipschitz continuity of B we have

$B\left(x,u,{\xi }^{\prime }\right)-B\left(x,\upsilon ,{\eta }^{\prime }\right)⩽b_1\left|{\xi }^{\prime }-{\eta }^{\prime }\right|+B\left(x,u,{\eta }^{\prime }\right)-B\left(x,\upsilon ,{\eta }^{\prime }\right),$

where b ₁ is the Lipschitz constant of B in the set K″. In particular, since B is non-increasing in z,

(3.5.6) $\begin{array}{cc}B\left(x,u,{\xi }^{\prime }\right)-B\left(x,\upsilon ,{\eta }^{\prime }\right)⩽b_1\left|{\xi }^{\prime }-{\eta }^{\prime }\right|& \text{when}u>\upsilon .\end{array}$

Step 2. By subtracting (3.5.1) and (3.5.2) we get

(3.5.7) $\text{div}\left\{A\left(x,Du\right)-A\left(x,D\upsilon \right)\right\}+B\left(x,u,Du\right)-B\left(x,\upsilon ,D\upsilon \right)⩾0$

in Ω. Let w = u − v and define

$\stackrel{\sim }{A}\left(x,\xi \right)=A\left(x,\xi +D\upsilon \left(x\right)\right)-A\left(x,D\upsilon \left(x\right)\right).$

Clearly

$\stackrel{\sim }{A}\left(x,Dw\right)=A\left(x,Du\right)-A\left(x,D\upsilon \right),$

so that in view of (3.5.7) the function w can be considered as a solution of the differential inequality

(3.5.8) $\text{div}\stackrel{\sim }{A}\left(x,Dw\right)+\stackrel{\sim }{B}\left(x,w,Dw\right)⩾0$

where $\stackrel{\sim }{B}\left(x,z,\xi \right)=B\left(x,z+\upsilon \left(x\right),\xi +D\upsilon \left(x\right)\right)-B\left(x,\upsilon \left(x\right),D\upsilon \left(x\right)\right)$ is defined analogously to Ã. Of course, also w = u − v ⩽ 0 on ∂Ω.

Since u, $\upsilon \in W_{\text{loc}}^{1,\infty }\left(\Omega \right)$ it follows that in any compact subset Ω′ of Ω we have Du, Dv ∈ R _W for some W > 0. Thus (3.5.5) and (3.5.6) hold in Ω′ with the identifications ξ′ = Du and η′ = Dv (so ξ′ − η′ = Dw); that is we have

$〈\stackrel{\sim }{A}\left(x,Dw\right),Dw〉⩾a_1{\left|Dw\right|}^2$

and

$\begin{array}{cc}\stackrel{\sim }{B}\left(x,w,Dw\right)⩽b_1\left|Dw\right|& \text{when}\end{array}w>\ell$

with ℓ ≥ Const. > 0. Stated in other terms, the functions à and $\stackrel{\sim }{B}$ in (3.5.8) obey the structural conditions (3.2.1) along the solution w, that is, with ξ = Dw and with also a ₂ = b ₂ = 0, p = 2.

Since w ⩽ 0 on ∂Ω we can therefore apply Theorem 3.2.1 to obtain w ⩽ 0 in Ω, that is u ⩽ v.

Remarks

This is essentially Theorem 10.7(i) of [40] with the important exceptions that A and B are allowed to be singular at ξ = 0, and that the class C ¹(Ω) is weakened to $W_{\text{loc}}^{1,\infty }\left(\Omega \right)$ . Compare also Theorem 10.3 of [71].

If Ω is unbounded and the boundary condition is understood to include the limit relation

$\underset{\left|x\right|\to \infty ,x\in \Omega }{\lim \sup }\left\{u\left(x\right)-\upsilon \left(x\right)\right\}⩽M,$

then the conclusion of Theorem 3.5.1 continues to hold. The same conclusion is valid for the later results of the section.

In the important case of the p-Laplace operator (where Q = {0}) we have the following corollary of Theorem 3.5.1.

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Inherently Parallel Algorithms in Feasibility and Optimization and their Applications

Sjur Didrik Flåm , in Studies in Computational Mathematics, 2001

7 REACHING EQUILIBRIUM

Motivated by format (10) as well as Propositions 2 and 4 we suggest that decentralized adaptations be modelled as follows. Iteratively, at stage k  =   0,1,… and for all i ϵ I, let

(17) $x_i^{k+1}=x_i^k+s_k\left\{\begin{array}{c}\hfill m_i\left(x^k\right)+{\lambda }_i\left[M_i\left(x^k\right)-M_i\left(x^{k-1}\right)\right]/s_{k-1}\mathit{if}{x}_i^k\in X_{i,}\hfill \\ -{\mathcal{N}}_i\left(x_i\right)\mathit{otherwise}.\hfill \end{array}\right\}$

For the convergence analysis we invoke henceforth the hypotheses of Proposition 2 and qualification (11).

Theorem 1

(Convergence of the discrete-time, heavy-ball method) Suppose M' is non-singular Lipschitz continuous on X, that set by assumption being bounded and containing only isolated solutions to (1). If the sequence {x^k } generated by (17) becomes feasible after finite time and {(x ^{k  +   1}-x^k )/s_k }remains bounded, then, provided all λ_i, are sufficiently large, x^k converges to a solution of (1).

Proof. We claim that $\omega \left\{\mathcal{X}\right\}$ , as defined in (15), is invariant under (10). Suppose it really is. Then, for some solution $\overline{x}$ to (1) we get $\omega \left\{\mathcal{X}\right\}=\left\{\overline{x}\right\}$ because singletons are the only invariant strict subsets of X (by Propositions 2 and 4). Thus, while assuming ω {χ} invariant, we have shown via Lemma 5 that $\omega \left\{x^k\right\}=\omega \left\{\mathcal{X}\right\}=\left\{\overline{x}\right\}$ whence x^k → $\overline{x}$ as desired.

To complete the proof we must verify that asserted invariance. That argument requires a detour. The Lipschitz continuity of M' (with modulus L) yields

$\begin{array}{l}M\left(x\right)-M\left(x^{-}\right)={\displaystyle \underset{0}{\overset{1}{\int }}}\frac{d}{dh}M\left(x^{-}+h\left(x-x^{-}\right)\right)dh\hfill \\ ={\displaystyle \underset{0}{\overset{1}{\int }}}{M}^{\prime }\left(x^{-}+h\left(x-x^{-}\right)\right)\left(x-x^{-}\right)dh\hfill \\ =M^{\prime }\left(x\right)\left(x-x^{-}\right)+{\displaystyle \underset{0}{\overset{1}{\int }}}\left[M^{\prime }\left(x^{-}+h\left(x-x^{-}\right)\right)-M^{\prime }\left(x\right)\right]\left(x-x^{-}\right)dh\hfill \\ \in M^{\prime }\left(x\right)\left(x-x^{-}\right)+\frac{L}{2}\Vert x-x^{-}\Vert {}^{2}B.\hfill \end{array}$

Therefore, letting x  = x^k , x ^-  = x ^k-1 , and ${\tau }_{k-\frac{1}{2}}:={\tau }_{k-1}+\frac{s_{k-1}}{2}$ we get via (13) that

$\begin{array}{l}\frac{M\left(x^k\right)-M\left(x^{k-1}\right)}{s_{k-1}}=M^{\prime }\left(x^k\right)\frac{x^k-x^{k-1}}{{\tau }_k-{\tau }_{k-1}}+O\left(s_k\right)=M^{\prime }\left(x^k\right)\stackrel{.}{\mathcal{X}}\left({\tau }_{k-\frac{1}{2}}\right)+O\left(s_k\right)\hfill \\ =M^{\prime }\left(\mathcal{X}\left({\tau }_{k-\frac{1}{2}}\right)\right)\stackrel{.}{\mathcal{X}}\left({\tau }_{k-\frac{1}{2}}\right)+O\left(s_k\right)=\stackrel{.}{M}\left(\mathcal{X}\left({\tau }_{k-\frac{1}{2}}\right)\right)+O\left(s_k\right).\hfill \end{array}$

As a result, again invoking (13), we see that when x^k ϵ X, system (17) can eventually be rewritten

(18) $\stackrel{.}{\mathcal{X}}\left(t\right)=m\left(\mathcal{X}\left({\tau }_k\right)\right)+\lambda \stackrel{.}{M}\left(\mathcal{X}\left({\tau }_{k-\frac{1}{2}}\right)\right)+O\left(s_k\right)\mathrm{for}t\in \left({\tau }_k,{\tau }_{k+1}\right).$

We now shift the initial time 0 of the process $\mathcal{X}$ backwards to gain from s_k   →   0⁺. More precisely, introduce a sequence of functions ${\mathcal{X}}^k:ℝ\to \mathbb{E},k=0,1,\dots$ defined by

(19) ${\mathcal{X}}^k\left(t\right):=\mathcal{X}\left({\tau }_k+t\right).$

(In particular, ${\mathcal{X}}^0=\mathcal{X}$ .) The sequence so constructed satisfies (16) for appropriate y^k , z^k ,r_k . To see this let the integer (counting) function

(20) $0\le r\mapsto \mathcal{K}\left(r\right):=\max \left\{k:{\tau }_k\le r\right\}.$

keep track of the stage. From (18) we get, writing $\widehat{k}=\mathcal{K}\left({\tau }_k+t\right)$ for short,

${\stackrel{.}{\mathcal{X}}}^k\left(t\right)=\frac{d}{dt}\mathcal{X}\left({\tau }_k+t\right)=m\left(\mathcal{X}\left({\tau }_{\widehat{k}}\right)\right)+\lambda \stackrel{.}{M}(\mathcal{X}\left({\tau }_{\widehat{k}-\frac{1}{2}}\right)+O\left(s_{\widehat{k}}\right)$

so (16) obtains with

$\begin{array}{l}{y}^k\left(t\right)=\mathcal{X}\left({\tau }_{\widehat{k}}\right)-{\mathcal{X}}^k\left(t\right)=x^{\mathcal{K}\left({\tau }_k+t\right)}-{\mathcal{X}}^k\left(t\right),\hfill \\ z^k\left(t\right)=\mathcal{X}\left({\tau }_{\widehat{k}-\frac{1}{2}}\right)-{\mathcal{X}}^k\left(t\right)=\mathcal{X}\left({\tau }_{\mathcal{K}\left({\tau }_k+t\right)-\frac{1}{2}}\right)-{\mathcal{X}}^k\left(t\right),\mathit{and}\hfill \\ r_n\left(t\right)=O\left(s_{\mathcal{K}\left({\tau }_{k+t}\right)}\right)\hfill \end{array}$

Since {x^k } is bounded, so are $\{{\mathcal{X}}^k(0)=x^k\}..\mathrm{and}.{\mathrm{\{z}}^k(0)\}$ . Clearly, r_k   →   0, and y^k   →   0 because $\Vert y^k\left(t\right)\Vert =\Vert x^{\mathcal{K}\left(\tau k+t\right)}-{\mathcal{X}}^k\left(t\right)\Vert \le \Vert x^{\mathcal{K}\left(\tau k+t\right)}-x^{\mathcal{K}\left(\tau k+t\right)+1}\Vert \to 0$ . Also, because

$\mathcal{X}\left({\tau }_{\mathcal{K}\left({\tau }_k+t\right)-\frac{1}{2}}\right)$ and X^k (t) both belong to the triangle conv $\left\{x^{\widehat{k}-1},x^{\widehat{k}},x^{\widehat{k}+1}\right\}$ , we get

$\begin{array}{l}\Vert z^k\left(t\right)\Vert =\Vert \mathcal{X}\left({\tau }_{\mathcal{K}\left({\tau }_k+t\right)-\frac{1}{2}}\right)-{\mathcal{X}}^k\left(t\right)\Vert \le \mathit{diam}\mathit{conv}\left\{x^{\widehat{k}-1},x^{\widehat{k}},x^{\widehat{k}+1}\right\}\hfill \\ \le \Vert x^{\widehat{k}-1}-x^{\widehat{k}}\Vert +\Vert x^{\widehat{k}}-x^{\widehat{k}+1}\Vert =O\left(s_{\widehat{k}-1}\right)+O\left(s_{\widehat{k}}\right)=O\left(s_{\widehat{k}-1}\right)\to 0.\hfill \end{array}$

When well defined ${\stackrel{.}{z}}^k\left(t\right)=-{\stackrel{.}{\mathcal{X}}}^k\left(t\right)=-\frac{d}{dt}\mathcal{X}\left({\tau }_k+t\right)$ . So, we also have

$\Vert {\stackrel{.}{z}}^k\left(t\right)\Vert =\Vert x^{\mathcal{K}\left({\tau }_k+t\right)+1}-x^{\mathcal{K}\left({\tau }_k+t\right)}\Vert /s_{\mathcal{K}\left({\tau }_k+t\right)}$

uniformly bounded by assumption.

After this detour we return to the claim that $\omega \left\{\mathcal{X}\right\}$ is invariant. Denote by ${\mathcal{X}}^k(t,x).$ and ${\mathcal{X}}^{\infty }(t,x).$ the unique state at time t of system (19) and (10) respectively provided it passes through x at time 0. Pick any point $x\in \omega \left\{\mathcal{X}\right\}.$ and any time t. We must show that ${\mathcal{X}}^{\infty }(t,x)\in \omega \left\{\mathcal{X}\right\}.$ . Via Lemma 5 some subsequence ${\{x^k=\mathcal{X}({\tau }_k)\}}_{k\in K}$ converges to x. If necessary, pass to a subsequence to have that ${\mathcal{X}}^k\to {\mathcal{X}}^{\infty }$ in the manner described in Lemma 6. Because ${\mathcal{X}}^{\infty }(0,x^k)={\mathcal{X}}^k(0,x^k)=x^k\to x$ as k ϵ K tends to infinity, this yields

(21) $\underset{k\in K}{lim}\Vert {\mathcal{X}}^{\infty }\left(t,x^k\right)-{\mathcal{X}}^k\left(t,x^k\right)\Vert =0.$

Therefore, since ${\mathcal{X}}^{\infty }(t,x^k)\to {\mathcal{X}}^{\infty }(t,x)$ by continuous dependence on initial conditions, (21) implies

${\mathcal{X}}^{\infty }\left(t,x\right)=\underset{k\in K}{lim}{\mathcal{X}}^{\infty }\left(t,x^k\right)=\underset{k\in K}{lim}{\mathcal{X}}^k\left(t,x^k\right)=\underset{k\in K}{lim}\mathcal{X}\left({\tau }_k+t\right)\in \omega \left\{\mathcal{X}\right\}.$

The asserted invariance has now been verified and this concludes the proof. ■

(17) is a purely primal procedure. Alternatively, using multipliers for penalization, one might develop primal-dual methods like those in [8].

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Measuring dependence and testing for independence

Dag Tjøstheim , ... Bård Støve , in Statistical Modeling Using Local Gaussian Approximation, 2022

7.5.1 Proof of Theorems 7.9 and 7.10

Proof of Theorem 7.9

Let $M<\infty$ and $\mathrm{\Delta }<\infty$ be such that $|L^{(\mathit{b})}((X_t,Y_t),\mathit{\theta })-L^{(\mathit{b})}((X_t,Y_t),{\mathit{\theta }}^{\prime })|⩽C_t|\mathit{\theta }-{\mathit{\theta }}^{\prime }|$ , $C_t⩽M$ , and, using 4-dominance, $|L^{(\mathit{b})}((X_t,Y_t),\mathit{\theta }){|}_4⩽\mathrm{\Delta }$ . For $\delta >0$ , let $\{\eta ({\mathit{\theta }}_i,\delta ):i=1,\dots ,B\}$ be a finite sub-cover of Θ, where $\eta ({\mathit{\theta }}_i,\delta )=\{\mathit{\theta }\in \mathbf{\Theta }:|\mathit{\theta }-{\mathit{\theta }}_i|<\delta \}$ . Then

(7.29) $\underset{\mathit{\theta }\in \mathbf{\Theta }}{\sup }|L_{n,\mathit{b}}(\mathit{\theta })-{\mu }_{\mathit{b}}|=\underset{i}{\max }\underset{\mathit{\theta }\in \eta ({\mathit{\theta }}_i,\delta )}{\sup }|L_{n,\mathit{b}}(\mathit{\theta })-{\mu }_{\mathit{b}}|.$

Hence we can write

(7.30) $P(\underset{\mathit{\theta }\in \mathbf{\Theta }}{\sup }|L_{n,\mathit{b}}(\mathit{\theta })-{\mu }_{\mathit{b}}|>ϵ)⩽\sum _{i=1}^{B}P(\underset{\mathit{\theta }\in \eta ({\mathit{\theta }}_i,\delta )}{\sup }|L_{n,\mathit{b}}(\mathit{\theta })-{\mu }_{\mathit{b}}|>ϵ).$

By the Lipschitz continuity of $\{L^{(\mathit{b})}((X_t,Y_t),\mathit{\theta })\}$ , if $\mathit{\theta }\in \eta ({\mathit{\theta }}_i,\delta )$ , then

$|L_{n,\mathit{b}}(\mathit{\theta })-{\mu }_{\mathit{b}}|⩽|L_{n,\mathit{b}}(\mathit{\theta })-L_{n,\mathit{b}}({\mathit{\theta }}_i)|+|L_{n,\mathit{b}}({\mathit{\theta }}_i)-{\mu }_{\mathit{b}}|⩽\frac{\delta }{n}\sum _{t=1}^n{C}_t+|L_{n,\mathit{b}}({\mathit{\theta }}_i)-{\mu }_{\mathit{b}}|.$

By the Markov inequality,

$P(\underset{\mathit{\theta }\in \eta ({\mathit{\theta }}_i,\delta )}{\sup }|L_{n,\mathit{b}}(\mathit{\theta })-{\mu }_{\mathit{b}}|>ϵ)⩽P(\frac{\delta }{n}\sum _{t=1}^n{C}_t>\frac{ϵ}{2})+P(|L_{n,\mathit{b}}({\mathit{\theta }}_i)-{\mu }_b|>\frac{ϵ}{2})⩽\frac{2\delta }{nϵ}\mathrm{\text{E}}\left(\sum _{t=1}^n{C}_t\right)+\frac{4}{{ϵ}^2}\text{Var}(L_{n,\mathit{b}}({\mathit{\theta }}_i))⩽\frac{2\delta M}{ϵ}+\frac{4}{{ϵ}^2}\text{Var}(L_{n,\mathit{b}}({\mathit{\theta }}_i)).$

Using the α-mixing assumption for $\{X_t,Y_t\}$ , by the mixing inequality (Corollary A.2, Hall and Heyde (1980)) and the 4-dominance of $\{L^{(\mathit{b})}((X_t,Y_t),\mathit{\theta })\}$ ,

$\text{Var}(L_{n,\mathit{b}}({\mathit{\theta }}_i))=\frac{1}{n^2}\text{Var}(\sum _{t=1}^n{L}^{(\mathit{b})}((X_t,Y_t),{\mathit{\theta }}_i))=\frac{1}{n^2}\sum _{t=1}^n\sum _{s=1}^n\text{Cov}(L^{(\mathit{b})}((X_t,Y_t),{\theta }_i),L^{(\mathit{b})}((X_s,Y_s),{\mathit{\theta }}_i))⩽\frac{8}{n^2}\sum _{t=1}^n\sum _{s=1}^n|L^{(\mathit{b})}((X_t,Y_t),{\mathit{\theta }}_i){|}_4|L^{(\mathit{b})}((X_s,Y_s),{\mathit{\theta }}_i){|}_4{\alpha }^{\frac{|t-s|}{2}}⩽\frac{8{\mathrm{\Delta }}^2}{n^2}\sum _{t=1}^n\sum _{s=1}^n{\alpha }^{\frac{|t-s|}{2}}=\frac{8{\mathrm{\Delta }}^2}{n^2}(n+2\sum _{j=1}^{n-1}(n-j){\alpha }^{\frac{j}{2}})⩽\frac{8{\mathrm{\Delta }}^2}{n^2}(n+2(n-1)\sum _{j=1}^{n-1}{\alpha }^{\frac{j}{2}})=\frac{8{\mathrm{\Delta }}^2}{n^2}(n+2(n-1)\frac{{\alpha }^{\frac{1}{2}}-{\alpha }^{\frac{n}{2}}}{1-{\alpha }^{\frac{1}{2}}}).$

Therefore

$P(\underset{\mathit{\theta }\in \eta ({\mathit{\theta }}_i,\delta )}{\sup }|L_{n,\mathit{b}}(\mathit{\theta })-{\mu }_{\mathit{b}}|>ϵ)⩽\frac{2\delta M}{ϵ}+\frac{32{\mathrm{\Delta }}^2}{n^2}(n+2(n-1)\frac{{\alpha }^{\frac{1}{2}}-{\alpha }^{\frac{n}{2}}}{1-{\alpha }^{\frac{1}{2}}})<\zeta +\frac{32{\mathrm{\Delta }}^2}{n^2}(n+2(n-1)\frac{{\alpha }^{\frac{1}{2}}-{\alpha }^{\frac{n}{2}}}{1-{\alpha }^{\frac{1}{2}}})$

for all n sufficiently large, $\zeta >0$ , and $\delta <\frac{ϵ\zeta }{2M}$ . From this it follows that, as $n\to \infty$ ,

$P(\underset{\mathit{\theta }\in \mathbf{\Theta }}{\sup }|L_{n,\mathit{b}}(\mathit{\theta })-{\mu }_{\mathit{b}}|>ϵ)<B\zeta ,\forall \zeta >0,B<\infty ,$

and therefore

$\underset{n\to \infty }{\lim }P(\underset{\mathit{\theta }\in \mathbf{\Theta }}{\sup }|L_{n,\mathit{b}}(\mathit{\theta })-{\mu }_{\mathit{b}}|>ϵ)=0,\forall ϵ>0.$

This means that

(7.31) $L_{n,\mathit{b}}(\mathit{\theta })-\mathrm{\text{E}}(L_{n,\mathit{b}}(\mathit{\theta }))\stackrel{P}{\to }0\text{uniformly on}\mathbf{\Theta }.$

Note that from the stationarity of the process $\{(X_t,Y_t)\}$ it follows that $\mathrm{\text{E}}(L_{n,\mathit{b}}(\mathit{\theta }))=\mathrm{\text{E}}(L^{(\mathit{b})}((X_t,Y_t),\mathit{\theta }))$ . Since $L_{n,\mathit{b}}$ is continuous on $\mathbf{\Theta }P$ -a.s. with maximizer ${\hat{\mathit{\theta }}}_{n,\mathit{b}}$ , from assumption B2 and (7.31) it follows that $L_{n,\mathit{b}}({\mathit{\theta }}_{0,\mathit{b}})-\mathrm{\text{E}}(L_{n,\mathit{b}}({\mathit{\theta }}_{0,\mathit{b}}))\stackrel{P}{\to }0$ . Moreover, using (7.31) and the definition of ${\hat{\mathit{\theta }}}_{n,\mathit{b}}$ and ${\mathit{\theta }}_{0,\mathit{b}}$ (that is, assumption B2), we have that $L_{n,\mathit{b}}({\hat{\mathit{\theta }}}_{n,\mathit{b}})-\mathrm{\text{E}}(L^{(\mathit{b})}((X_t,Y_t),{\mathit{\theta }}_{0,\mathit{b}}))={\sup }_{\mathit{\theta }\in \mathbf{\Theta }}{L}_{n,\mathit{b}}(\mathit{\theta })-{\sup }_{\mathit{\theta }\in \mathbf{\Theta }}\mathrm{\text{E}}(L^{(\mathit{b})}((X_t,Y_t),\mathit{\theta }))\stackrel{P}{\to }0$ . This means that $L_{n,\mathit{b}}({\hat{\mathit{\theta }}}_{n,\mathit{b}})-L_{n,\mathit{b}}({\mathit{\theta }}_{0,\mathit{b}})\stackrel{P}{\to }0$ . By the assumption that ${\hat{\mathit{\theta }}}_{n,\mathit{b}}$ is a maximizer of $L_{n,\mathit{b}}$ , for every $ϵ>0$ , there exists $\eta >0$ such that $|L_{n,\mathit{b}}({\hat{\mathit{\theta }}}_{n,\mathit{b}})-L_{n,\mathit{b}}(\mathit{\theta })|>\eta$ for every θ with $|{\mathit{\theta }}_{n,\mathit{b}}-\mathit{\theta }|>ϵ$ . Therefore, if we take $\mathit{\theta }={\mathit{\theta }}_{0,\mathit{b}}$ , then the event $\{|{\hat{\mathit{\theta }}}_{n,\mathit{b}}-{\mathit{\theta }}_{0,\mathit{b}}|>ϵ\}$ is contained in the event $\{|L_{n,\mathit{b}}({\hat{\mathit{\theta }}}_{n,\mathit{b}})-L_{n,\mathit{b}}({\mathit{\theta }}_{0,\mathit{b}})|>\eta \}$ , meaning that for every $ϵ>0$ , there exists $\eta >0$ such that

$P(|{\hat{\mathit{\theta }}}_{n,\mathit{b}}-{\mathit{\theta }}_{0,\mathit{b}}|>ϵ)⩽P(|L_{n,\mathit{b}}({\hat{\mathit{\theta }}}_{n,\mathit{b}})-L_{n,\mathit{b}}({\mathit{\theta }}_{0,\mathit{b}})|>\eta )\stackrel{P}{\to }0.$

The last statement of the theorem follows since ${\mathit{\theta }}_{0,\mathit{b}}\to {\mathit{\theta }}_0$ by definition.   □

Proof of Theorem 7.10

It is a generalization of Theorem 3 of Tjøstheim and Hufthammer (2013) and of the proof of Theorem 4.2. Define $Q_n(\mathit{\theta })=-\frac{n}{{(b_1{b}_2)}^2}{L}_{n,\mathit{b}}(\mathit{\theta })$ and consider the Taylor expansion of $\mathrm{\nabla }{Q}_n(\mathit{\theta })$ :

$0=\frac{1}{\sqrt{n}}\mathrm{\nabla }{Q}_n({\hat{\mathit{\theta }}}_{n,\mathit{b}})=\frac{1}{\sqrt{n}}\mathrm{\nabla }{Q}_n({\mathit{\theta }}_{0,\mathit{b}})+\frac{1}{n}{\mathrm{\nabla }}^2{Q}_n(\tilde{\mathit{\theta }})\sqrt{n}({\hat{\mathit{\theta }}}_{n,\mathit{b}}-{\mathit{\theta }}_{0,\mathit{b}}),$

where $\tilde{\mathit{\theta }}$ is determined by the mean value theorem. Therefore

$-\frac{{(b_1{b}_2)}^{\frac{3}{2}}}{\sqrt{n}}\mathrm{\nabla }{Q}_n({\mathit{\theta }}_{0,\mathit{b}})=\frac{{(b_1{b}_2)}^{\frac{3}{2}}}{n}{\mathrm{\nabla }}^2{Q}_n(\tilde{\mathit{\theta }})\sqrt{n}({\hat{\mathit{\theta }}}_{n,\mathit{b}}-{\mathit{\theta }}_{0,\mathit{b}})=\frac{{(b_1{b}_2)}^{\frac{3}{2}}}{n}[{\mathrm{\nabla }}^2{Q}_n({\mathit{\theta }}_{0,\mathit{b}})+({\mathrm{\nabla }}^2{Q}_n(\tilde{\mathit{\theta }})-{\mathrm{\nabla }}^2{Q}_n({\mathit{\theta }}_{0,\mathit{b}}))]\sqrt{n}({\hat{\mathit{\theta }}}_{n,\mathit{b}}-{\mathit{\theta }}_{0,\mathit{b}}).$

Suppose we can prove that

(i)

$\frac{1}{n}\mathrm{\nabla }{Q}_n({\mathit{\theta }}_{0,\mathit{b}})\to 0$ a.s.;

(ii)

$\frac{1}{n}{\mathrm{\nabla }}^2{Q}_n({\mathit{\theta }}_{0,\mathit{b}})\to \tilde{\mathit{J}}$ a.s., where $\tilde{\mathit{J}}$ is a $5\times 5$ positive definite matrix that can be identified with the limit of $\frac{1}{{(b_1{b}_2)}^2}{\mathit{J}}_{n,\mathit{b}}$ as $n\to \infty$ and $\mathit{b}\to 0$ ;

(iii)

${\lim }_{n\to \infty }{\lim \sup }_{\delta \to 0}\frac{1}{n\delta }|{\mathrm{\nabla }}^2{Q}_n(\tilde{\mathit{\theta }})-{\mathrm{\nabla }}^2{Q}_n({\mathit{\theta }}_{0,\mathit{b}})|<\infty$ ;

(iv)

$\text{Var}(-\frac{{(b_1{b}_2)}^{\frac{3}{2}}}{\sqrt{n}}\mathrm{\nabla }{Q}_n({\mathit{\theta }}_{0,\mathit{b}}))=\frac{1}{{(b_1{b}_2)}^2}{\mathit{M}}_{n,\mathit{b}}$ .

Then, using Theorem 4.4 of Masry and Tjøstheim (1995) and Theorem 2.2 of Klimko and Nelson (1978), we have the result.

The proof of (i)–(iii) is essentially the same as that of Theorem 4.2.

Finally, item (iv) is a straightforward consequence of the definition of $\mathrm{\nabla }{Q}_n({\mathit{\theta }}_{0,\mathit{b}})$ .   □

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